Fitting Smooth Histories to Rotation Data

Consider two tectonic plates diverging at a mid-ocean ridge. Geophysicists are able to estimate the rotation of one plate relative to the other at a discrete sequence of times in the earth's history; also they usually have information as to the likely errors in these rotation estimates. We address the problem of fitting a smooth history to such rotation data. We employ a modification of the method used by Jupp and Kent in their 1987 article dealing with fitting a smooth history to time-labeled points on the surface of the unit sphere in three-dimensional space. They use parallel translation to "unroll" data from the surface of the sphere to a plane. We replace unrolling via parallel translation by unrolling via left group multiplication, using the group structure of SO(3). We explain why our understanding of the errors in tectonic plate reconstructions dictates that left group multiplication is preferable both to parallel translation and to right group multiplication. To choose the smoothing parameter we use the discrepancy method; for the Central Atlantic data set which we consider this method gives considerably better results than cross-validation.